# Questions tagged [chromatic-polynomial]

The chromatic-polynomial tag has no usage guidance.

25
questions

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### Sum of squares of chromatic roots of a bipartite graph

Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...

5
votes

0
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275
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### Which coefficient of a chromatic polynomial is the largest?

Let $\chi_G(q)$ be the chromatic polynomial of a graph $G$ with $n$
vertices. (More generally, $\chi_{\mathcal{A}}(q)$ can be the
characteristic polynomial of a finite hyperplane arrangement
$\mathcal{...

4
votes

0
answers

122
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### Chromatic number of rectangle tilings

Suppose we have a region of the plane tiled by finitely many
rectangles. We want to color the rectangles so that two
rectangles have different colors if they share a part of an
edge or if they share ...

0
votes

0
answers

190
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### Computational practicality of proving a theorem by transforming into a map coloring and finding $P(3)$, where $P$ is the chromatic polynomial

So I have heard that you can transform a math statement into a map in such a way that proving the statement is true is equivalent to finding a $3$-coloring of that map.
I'm also aware that you can ...

6
votes

1
answer

171
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### On a limit involving a transform of the chromatic polynomial

I was playing around with the chromatic polynomial (denoted here by $\chi_G(x)$) and I have made the following conjecture.
Let $(G_n)_{n \ge 1}$ be a sequence of graphs with $v(G_n) \to \infty$ ($v(...

1
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0
answers

124
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### The chromatic polynomial of a line graph

Is there a way to obtain the chromatic polynomial of the line graph of a regular simple graph, having known the chromatic polynomial of the graph?
There already exist characterizations of line graph ...

2
votes

0
answers

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### chromatic class of graphs of order $n$

Let $\mathcal{G}(n)$ be the isomorphism class of simple graphs of order $n$. We say two graphs in $\mathcal{G}(n)$ are chromatic equivalent if their chromatic polynomials have an equal linear ...

0
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1
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129
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### Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points [closed]

Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...

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444
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### Chromatic polynomial of a bipartite graph replaced by a new graph

Consider a semi-regular bipartite graph $C$ consisting of two parts $A$ (having each vertex of degree $\Delta$) and $B$ (having each vertex of degree $2$). Let its chromatic polynomial be $C(x)$. Now,...

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0
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123
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### Linear coefficient of chromatic polynomial

I am interested in the combinatorics of the linear coefficient of the chromatic polynomials. I have the following questions.
What are some class of graphs for which it is possible to calculate this ...

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0
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104
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### Bounds on spectral radius using chromatic number

I am struggling with this question:
If I have a connected graph $G$ on $n$ vertices and $m$ edges with chromatic number $d$ then how can I give a bound(lower and upper) on its spectral radius in ...

7
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### Chromatic polynomial and the circle

In https://arxiv.org/pdf/1208.5781.pdf
It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$.
My ...

1
vote

1
answer

141
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### Extension of chromatic polynomial to multi graphs

Let $G$ be a multigraph, i.e, there can be more than one edge between a pair of vertices. It is clear that the chromatic polynomial cannot capture these multi-edges. Because chromatic polynomial just ...

0
votes

1
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122
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### Chromatic Polynomials of Circulant Graph With Two Parameters

I have been working with the chromatic polynomials of circulant graphs of prime order $p$ with two distinct parameters, i.e.
$P_{p,i,j}(x):= P(C_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$
In ...

0
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1
answer

357
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### Chromatic polynomial for hyper cube [closed]

Does anyone know the chromatic polynomial of the hyper cube graph Q4?
I need this to verify that my listing of a subset of all DAG's on the 4-cube is correct.
Any help greatly appreciated,
JC

0
votes

0
answers

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### Expressions for the chromatic polynomial of a graph G

Chromatic polynomial of a graph $G$ is an important tool in Graph theory which has been studied extensively from graph theory perspective as well as through other area of Mathematics also. Hence it is ...

11
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2
answers

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### How many chromatic polynomials of planar maps are there?

Let P(n) be the set of polynomials that can occur as the chromatic polynomial of a planar map with n countries. What is known or conjectured about the growth of |P(n)|?
PS: Thanks Gerry and Noam, ...

3
votes

1
answer

731
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### chromatic polynomial of G - Join graph

Given a connected graph $G$ with $n$ vertices and given set of $\{m_1,m_2,...,m_n\}$ $n$ integers, we form a new graph $G^{\wedge} $ by considering the complete graph $K_{m_i}$ for each vertex i and '...

6
votes

0
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### Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then $$\...

9
votes

1
answer

476
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### Has anyone seen this sort of graph property used before?

Consider the following property of a graph $G$:
The graph $G$ has no independent cutset of vertices, $S$, such that the number of components of $G-S$ is more than $|S|$ (the size of $S$).
(That is, ...

3
votes

1
answer

757
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### Graphical representation of chromatic polynomial

Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that
$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle.
Also, we know ...

1
vote

1
answer

129
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### Non-alternating chromatic factors?

It is well-known that the coefficients of a chromatic polynomial alternate in sign. But is it possible for a chromatic polynomial to have a factor (over $\mathbb{Q}$) with coefficients which do not ...

5
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2
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### polynomials with the same discriminant

Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$.
I have ...

11
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2
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### Graphs with the same chromatic symmetric function

Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s [http://math.mit.edu/~...

13
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0
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### Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...